#7868932

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,15,16,14)(17,18,23,24)(19,22,27,32)(20,25,29,31)(21,30,28,26);(1,3,6,5)(2,7,4,8)(9,11,13,16)(10,14,12,15)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,26,31,30);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,16,15)(18,24)(19,27)(20,29)(26,30);(9,10,13,12)(11,14,16,15)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,31)(30,32);(1,4,6,2)(3,8,5,7)(9,10,13,12)(11,14,16,15)(17,21,23,28)(18,26,24,30)(19,20,27,29)(22,31,32,25);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,22)(18,27)(19,24)(20,26)(21,25)(23,32)(28,31)(29,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7776128);